Form-Invariance of the Non-Regular Exponential Family of Distributions Distribuciones de forma invariante de la familia exponencial no regular

The weighted distributions are used when the sampling mechanism records observations according to a nonnegative weight function. Sometimes the form of the weighted distribution is the same as the original distribution except possibly for a change in the parameters that is called the form-invariant weighted distribution. In this paper, by identifying a general class of weight functions, we introduce an extended class of form-invariant weighted distributions belonging to the non-regular exponential family which included two common families of distribution: exponential family and non-regular family as special cases. Some properties of this class of distributions such as the su cient and minimal su cient statistics, maximum likelihood estimation and the Fisher information matrix are studied.


Introduction
The weighted distributions have been used when the sampling mechanism records observations according to a certain chance.For example, suppose that the random variable X has the probability density function (pdf) f (x; θ) with parameter θ ∈ Θ and that the probability of the recording observation x of X be proportional to a non-negative weight function w(x, θ, β) where β is a known or unknown parameter, and that assumed E w(X, θ, β) exists.Then the recorded x is an observation of X w , having a pdf f w (x; θ, β) = w(x, θ, β)f (x; θ) E w(X, θ, β) . (1) The weighted distributions are widely applied in various studies such as line transect sampling, renewal theory, branching process, bio-medicine, statistical ecology and reliability modeling.Some references are Patil & Rao (1978); Rao (1958); Gupta & Kirmani (1990); Oluyede & George (2002) and Gupta & Keating (1986).
An important property of weighted distributions is that the original distribution keeps the same form under weighting except possibly for a change in the parameters.This is known as the form-invariant property of weighted distributions.According to Patil & Ord (1976), the distribution of X is said to be form-invariant under the weight function w(x, θ) when where η = η(θ) ∈ Θ.The form-invariance is a useful property for the weighted distributions because the estimate of the population parameters based on a random variable could be used in weighted cases, with considerable modication (Nair & Sunoj (2003)).Patil & Ord (1976) proved that a necessary and sucient condition for X to be form-invariant under size-biased weight function of order β (i.e.w(x, β) = x β ) is that its distribution belongs to log-exponential family with pdf f (x; θ) = exp{θ log x + a(x) − c(θ)}.Sankaran & Nair (1993) and Sindu (2002), respectively, derived the conditions which the Pearson and generalized Pearson families of distributions are forminvariant under w(x) = x.Alavi & Chinipardaz (2009) developed the Patil & Ord (1976) results and gave the necessary and sucient conditions for the forminvariance of the distributions belong to exponential family with pdf under the general weight function w β) .Esparza (2013) proved that two discrete distribution phase-type and matrix-geometric distributions are form-invariant under the size-biased factorial sampling of order β (i.e.w(x, β) = There are the distributions that their pdf form can be written as the function form of (3) and also are form-invariant under the weight function w(x, β) = [ν(x)] h(β) but do not belong to the exponential family.For example, consider the exponential distribution with two unknown parameters of α and θ and the pdf but does not belong to exponential family.
In this paper, we study the form-invariant property of the extended class of distributions, termed as the non-regular exponential family with pdf, given by where π 1 (θ) < π 2 (θ) for all θ ∈ Θ.The proposed class of the distributions include two common families of distributions; non-regular family and the exponential family, as particular cases.We identied the class of the weight functions and proved that a necessary and sucient condition for a distribution to be form-invariant under w ∈ W is that its pdf is the form (4).
For further insights on the form-invariance, we have extended the study to the two-parameter distributions that belong to the non-regular exponential family and proved the necessary and sucient conditions for the form-invariance property.Finally, some properties of the distributions belong to the nonregular-exponential family such as the sucient and minimal sucient statistics, maximum likelihood estimation and the Fisher information matrix are given.

Form-Invariance in One-Parameter Distributions
In this Section, we extend the Patil & Ord (1976) and Alavi & Chinipardaz (2009) results for a larger class of distributions such as nonregular-exponential family.
Theorem 1. vet X e rndom vrile tht hnges over n open intervl e lss of weight funtions whih for ll θ ∈ ΘF uppose the following onditions re stis(edY whih c(θ) = log Proof .According to the dinition of form-invarince in (2), it is clear that for each f , η is a function of β with η = θ when β = 0 and Therefore, it can be said that η = θ if and only if . Suppose that X has the pdf (5), then , Also, we have Revista Colombiana de Estadística 41 (2018) 157172 and By an application of Dominated Convergence Theorem (see Billingsley 1979) and the Hospital's rule, the equation ( 6) is as which the notation `DCT ' in the proof is abbrivation of the Domiated Convergence Theoerm.To prove the suciency, we note that Taking limit of both sides as η → θ and using the conditions in theorem 1, so that where b Now, integrating with respect to θ in (8), we get where B 1 (θ) = b 1 (θ) dθ, C 1 (θ) = b 2 (θ) dθ and a(x) is the constant coecient of integration.Thus the pdf of X must be the form (5).
Corollary 1. hen the funtions of π 1 nd π 2 in (5) re independent of θD the resultnt distriution is elong to exponentil fmilyF sn this seD X is formE invrint under the weight funtion w(x; β) = exp{β d(x) if nd only if its pdf hs the form (3)F This is studied by Patil & Ord (1976) and Alavi & Chinipardaz (2009) by dening, respectively, d(x) = log ν(x) and d(x) = log x.
Corollary 2. he nonEregulr fmily of distriutions with the pdf s is speil se of (5) when d(x) = 0F sn this seD X is formEinvrint under the weight funtion w(x, θ) = Some examples of the form-invariant one-parameter distributions belong to the non-regular exponential family with the common weight functions are given in Table 1.

Form-Invariance in Two-Parameter Distributions
The pdf of the multi-parameter distributions in nonregular exponential family is dened as where θ ∈ θ is a vector of parameters.By reparametrizing of (9), as α i = b i (θ) and p = 2, the pdf of two-parameter distributions in nonregular exponential family is written as (11) Alavi & Chinipardaz (2009) gave some examples of the form-invariant two-parameter distributions belong to exponential family but they could not give a general expression.In this Section, we show that the form-invariant property is extensible for all two-parameter distributions belong to non-regular exponential family.Normal (µ, σ) Theorem 2. vet the rndom vrile XD with the pdf f (x; α)D hnges over n open intervl I = π 1 (α), π 2 (α) F elso let e lss of weight funtions whih henD the distriution of X is formEinvrint under w ∈ W if nd only if the pdf is the form of (10)F Proof .According (2), for each f , η = α if and only if (β 1 , β 2 ) = (0, 0) and where c w (η) = log For example for i = 1, the equation ( 13) is Revista Colombiana de Estadística 41 (2018) 157172 To prove the suciency, we note that Now, when η 1 → α 1 in (14), by using the conditions (a) and (b) in theorem, we have where By integrating to α 1 , we get where is the constant of integration.Thus, Similarly, when From ( 16) and (17), Since ( 18) holds for all α 1 and α 2 , both sides must be of the form a(x) and independent of α.Thus the density must be the form as (10).
Some examples of the form-invariant two-parameter distributions belong to the non-regular exponential family with the common weight functions are given in Table 2.
Table 2: Form-invariant two-parameter distributions in non-regular exponential family.
The shapes of some common form-invariant distributions in the nonregular exponential family with their weighted versions are given in Figure 1.

Some Properties
In this Section, some properties of the form-invariant two-parameter distributions with pdf (10) are derived.
(a) The sucient and minimal sucient statistics under the random sampling and also the form-invariant weighted sampling of size n have the same form, respectively T (X) and T (X w ), as and Figure 1: Some form-invariant distributions in nonregular exponential family.
Example 1.Let X have the exponential distribution with pdf as ) are sucient, minimal sucient and also complete statistics for (α, θ) in (19) under the random sampling and the weighted sampling with the weight function e βx , respectively.
Example 2. Suppose that x is a random sample from a population with the pdf as f (x; α) = c(α)h(x) I (0,α) (x), where c(.) and h(.) are positive functions.Also, let x w is a weighted version of x under the weight function w(x, θ, β) = I (0,βα) (x) where β ∈ (0, 1) is known.The likelihood functions in x and x w is calculated, respectively, as and 2 < 0, thus the functions c(α), c(βα), L(α) and L w (α) are decreasing functions with respect to α.Therefore, the maximum likelihood estimator for α under the random sampling and the weighted samples are, respectively, X (n) and Revista Colombiana de Estadística 41 (2018) 157172 (c) Suppose that {P α } non-regular exponential family with the pdf of the form (10).The partial derivatives of any order can be obtained by dierentiating inside the integral sign.
Now, let the functions π 1 and π 2 is satised in following conditions then, according to the regularity condition of Rao (1965), the Fisher information matrix on α in f , I, and also in its weighted version (f w ), I w , is calculated as According to Patil & Taillie (1987), an intrinsic comparison between the weighted observations and original observations is possible when the dierence of the matrices, I w (α) − I(α), is either positive denite or negative denite.Positive (negative) deniteness of the dierence matrix means that every scalar-valued function of α can be estimated with smaller (larger) asymptotic standard error under the weighted observations.Therefore, the weighted observations can be or not favorable.When I w (α) − I(α) is indenite, comparison of the weighted and the original observations can be made in terms of the comparison of scalar-valued measure of the generalized variance, i.e., the determinant of the Fisher information matrix.Based on this measure, the observations (X or X w ) that have the smallest generalized variance will be more favored.

Conclusion
In the paper, we studied the problem of form-invariance in the non-regular exponential family when the original distribution is subjected to a weighted distribution.The class includes many common distributions, such as Patil & Ord (1976) and Billingsley (1979).It was shown that the maximum likelihood estimator could be obtained dierently than original ones.The Fisher information matrix for weighted distribution is compared with original distribution to show which one is more (less) informative.

Table 1 :
Form-invariant one-parameter distributions in non-regular exponential family.